Smile and Term Structure Revisited¶
One of the most visible failures of the Black–Scholes model is the presence of a volatility smile and a non-flat term structure of implied volatility. These phenomena summarize how markets deviate from constant-volatility assumptions and encode rich information about risk-neutral distributions.
Cross-reference: This section builds on the implied volatility concepts developed in Chapter 7. Here we focus on how these phenomena motivate stochastic volatility models.
The Implied Volatility Surface¶
Given a market price \(C^{\text{mkt}}(K, T)\) for a European call option, the implied volatility \(\sigma_{\text{impl}}(K, T)\) is defined as the unique value solving:
If the Black–Scholes model were correct, \(\sigma_{\text{impl}}(K, T) = \sigma\) would be constant. Instead, market data reveal a two-dimensional surface with systematic patterns.
Parameterizations:
Rather than strike \(K\), practitioners often use: - Log-moneyness: \(k = \log(K/F)\) where \(F = S_0 e^{(r-q)T}\) is the forward - Delta: The Black–Scholes delta of the option - Standardized moneyness: \(m = k/(\sigma_{\text{ATM}}\sqrt{T})\)
Smile Patterns¶
Equity Index Smile (Skew)¶
For equity indices (S&P 500, Euro Stoxx 50), the dominant feature is negative skew:
Typical characteristics:
| Feature | S&P 500 (typical) |
|---|---|
| ATM implied vol | 15%–20% |
| 25-delta put vol | ATM + 3% to 8% |
| 25-delta call vol | ATM − 0.5% to 1% |
| Skew (per 10% moneyness) | 1.5%–4% |
The asymmetry reflects: 1. Crash risk: Markets assign higher probability to large downward moves 2. Leverage effect: Volatility increases when prices fall 3. Demand imbalance: Strong demand for downside protection (puts)
Single-Stock Smiles¶
Individual equities exhibit more varied patterns: - Generally less steep skew than indices - Can show upward-sloping wings for speculative stocks - Earnings and events create localized smile distortions
FX Smiles¶
Currency options typically show a more symmetric smile (not skew):
Both OTM puts and OTM calls have elevated implied volatility, reflecting tail risk in both directions.
Commodity Smiles¶
Commodities exhibit varied patterns depending on: - Supply/demand asymmetries - Storage costs and convenience yields - Seasonality
Term Structure of Implied Volatility¶
Implied volatility varies systematically with maturity:
Common Term Structure Shapes¶
Upward sloping (contango): Short-term vol < Long-term vol - Typical in calm markets - Reflects uncertainty increasing with horizon - Mean reversion of volatility is not yet "priced in"
Downward sloping (backwardation): Short-term vol > Long-term vol - Common during stress periods - Current high volatility expected to mean-revert - VIX term structure in 2008, 2020 exhibited strong backwardation
Humped: Peak at intermediate maturity - Event risk (earnings, elections) at specific dates - Regulatory deadlines or macro announcements
Quantitative Relationships¶
For ATM options, empirical regularities include:
Variance additivity (approximately):
Mean reversion signature: Under a mean-reverting volatility process with speed \(\kappa\) and long-run level \(\bar{\sigma}\):
This interpolates between \(V_0\) (short term) and \(\bar{\sigma}^2\) (long term).
Joint Smile-Term Structure Dynamics¶
The smile and term structure interact in important ways:
Smile Flattening with Maturity¶
Short-maturity smiles are typically steeper than long-maturity smiles:
Scaling relationship: Under diffusion models, the skew scales approximately as:
Smile Dynamics¶
As the underlying moves, the smile shifts. Two limiting cases:
Sticky strike: \(\sigma_{\text{impl}}(K, t)\) remains constant for fixed \(K\) - Implies: as \(S\) moves, the smile "slides" along the strike axis - Inconsistent with most stochastic volatility models
Sticky delta: \(\sigma_{\text{impl}}(\Delta, t)\) remains constant for fixed \(\Delta\) - Implies: the smile moves with the forward - More consistent with floating smile models
Empirically, reality lies between these extremes and varies by market regime.
Economic Interpretation¶
The implied volatility surface encodes market beliefs and risk premia:
Risk-Neutral Density¶
The Breeden–Litzenberger formula (Chapter 7) recovers the risk-neutral density:
A non-flat smile implies a non-Gaussian risk-neutral density: - Steep put skew → heavy left tail → crash risk priced - Elevated wings → fat tails → extreme moves priced
Variance Risk Premium¶
The gap between implied and realized variance:
is typically positive for equity indices (investors pay a premium for volatility protection). The term structure of VRP reveals: - Short-term VRP tends to be larger - VRP increases during stress - Mean reversion of VRP itself
Demand and Supply¶
The smile also reflects market microstructure: - Institutional demand for OTM puts (portfolio insurance) - Market makers' inventory and hedging costs - Flow imbalances in specific strikes/maturities
Implications for Stochastic Volatility Models¶
A viable stochastic volatility model must explain:
| Feature | Model requirement |
|---|---|
| Smile existence | Non-constant instantaneous volatility |
| Negative equity skew | Negative price-vol correlation (\(\rho < 0\)) |
| Smile curvature | Volatility of volatility (\(\xi > 0\)) |
| Skew decay with \(T\) | Mean reversion of volatility |
| Term structure | Realistic volatility dynamics |
The Heston model (Section 9.3) addresses all these through: - Stochastic variance \(V_t\) - Correlation \(\rho\) (skew) - Vol-of-vol \(\xi\) (curvature) - Mean reversion \(\kappa\) (term structure)
Empirical Smile: S&P 500 Example¶
Representative 1-month smile (normal conditions):
| Strike (% of spot) | Implied Vol |
|---|---|
| 90% | 22% |
| 95% | 18% |
| 100% (ATM) | 15% |
| 105% | 14% |
| 110% | 13.5% |
Representative term structure (ATM):
| Maturity | Implied Vol |
|---|---|
| 1 week | 14% |
| 1 month | 15% |
| 3 months | 16% |
| 6 months | 17% |
| 1 year | 18% |
| 2 years | 19% |
These patterns vary with market conditions but the qualitative features persist.
Key Takeaways¶
- The volatility smile is ubiquitous and persistent across asset classes
- Equity indices exhibit negative skew; FX shows more symmetric smiles
- Term structure reflects volatility mean reversion and risk premia
- Short-maturity smiles are steeper than long-maturity smiles
- The smile encodes risk-neutral beliefs about tail risk
- Any realistic model must explain smile, skew, and term structure jointly
Further Reading¶
- Rubinstein, M. (1994). Implied binomial trees. Journal of Finance.
- Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.
- Carr, P. & Wu, L. (2009). Variance risk premiums. Review of Financial Studies.
- Bates, D. (2000). Post-'87 crash fears in the S&P 500 futures options market. Journal of Econometrics.
- Cont, R. & da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quantitative Finance.
Exercises¶
Exercise 1. The following ATM implied volatilities are observed for S&P 500 options:
| Maturity | Implied Vol |
|---|---|
| 1 month | 22% |
| 3 months | 19% |
| 6 months | 17% |
| 1 year | 16% |
Is the term structure in contango or backwardation? What does this shape suggest about the current volatility regime relative to the long-run level? Estimate the long-run volatility \(\bar{\sigma}\) implied by the data.
Solution to Exercise 1
Contango vs. backwardation: The term structure is in backwardation (also called inverted). Short-term implied volatility (22%) exceeds long-term implied volatility (16%), meaning \(\sigma_{\text{impl}}(T)\) is decreasing in \(T\).
Current volatility regime: Backwardation indicates that current (instantaneous) volatility is above the long-run equilibrium level. The market expects volatility to mean-revert downward over time. This shape is typical during or just after a period of market stress, when realized volatility is elevated but expected to normalize.
Estimating long-run volatility \(\bar{\sigma}\): Under a mean-reverting volatility process, the implied volatility for very long maturities converges to \(\bar{\sigma}\). The data show a decreasing pattern:
| Maturity | Implied Vol |
|---|---|
| 1 month | 22% |
| 3 months | 19% |
| 6 months | 17% |
| 1 year | 16% |
The rate of decrease is slowing (3% drop from 1m to 3m, 2% from 3m to 6m, 1% from 6m to 1y), suggesting convergence. Extrapolating, \(\bar{\sigma} \approx 15\%\)–\(16\%\). A reasonable estimate is \(\bar{\sigma} \approx 15\%\), representing the level the term structure is asymptotically approaching.
Exercise 2. Under a mean-reverting volatility process, the ATM implied variance satisfies approximately
Given \(V_0 = (25\%)^2\), \(\bar{\sigma}^2 = (18\%)^2\), and \(\kappa = 2.0\), compute \(\sigma_{\text{impl}}(T)\) for \(T = 0.25, 0.5, 1.0, 2.0\) years. Verify that the term structure interpolates between \(\sqrt{V_0}\) and \(\bar{\sigma}\).
Solution to Exercise 2
The formula is
with \(V_0 = 0.0625\), \(\bar{\sigma}^2 = 0.0324\), and \(\kappa = 2.0\).
Define \(g(T) = \dfrac{1 - e^{-\kappa T}}{\kappa T}\), so \(\sigma_{\text{impl}}^2(T) = V_0 \cdot g(T) + \bar{\sigma}^2 \cdot (1 - g(T))\).
For \(T = 0.25\): \(g(0.25) = \dfrac{1 - e^{-0.5}}{0.5} = \dfrac{1 - 0.6065}{0.5} = \dfrac{0.3935}{0.5} = 0.7870\)
For \(T = 0.5\): \(g(0.5) = \dfrac{1 - e^{-1.0}}{1.0} = 1 - 0.3679 = 0.6321\)
For \(T = 1.0\): \(g(1.0) = \dfrac{1 - e^{-2.0}}{2.0} = \dfrac{1 - 0.1353}{2.0} = \dfrac{0.8647}{2.0} = 0.4324\)
For \(T = 2.0\): \(g(2.0) = \dfrac{1 - e^{-4.0}}{4.0} = \dfrac{1 - 0.01832}{4.0} = \dfrac{0.9817}{4.0} = 0.2454\)
Summary:
| \(T\) | \(g(T)\) | \(\sigma_{\text{impl}}\) |
|---|---|---|
| 0.25 | 0.787 | 23.68% |
| 0.50 | 0.632 | 22.68% |
| 1.00 | 0.432 | 21.31% |
| 2.00 | 0.245 | 19.95% |
Verification: As \(T \to 0\), \(g(T) \to 1\), so \(\sigma_{\text{impl}} \to \sqrt{V_0} = 25\%\). As \(T \to \infty\), \(g(T) \to 0\), so \(\sigma_{\text{impl}} \to \bar{\sigma} = 18\%\). The computed values decrease monotonically from near 25% toward 18%, confirming the interpolation property.
Exercise 3. For equity index options, the implied volatility skew is often measured as
where percentages refer to strike as a fraction of spot. Suppose the 1-month skew is 7% and the 1-year skew is 2.5%. Verify that this is roughly consistent with the scaling relationship \(\text{Skew}(T) \propto 1/\sqrt{T}\). What mechanism in stochastic volatility models generates this decay?
Solution to Exercise 3
Verifying the scaling relationship:
If \(\text{Skew}(T) \propto 1/\sqrt{T}\), then
With \(T_1 = 1/12\) year (1 month) and \(T_2 = 1\) year:
The observed ratio:
The predicted ratio is 3.46, while the observed ratio is 2.80. These are in rough agreement (within the range expected given the approximation), confirming the \(1/\sqrt{T}\) scaling as a reasonable first-order description. The slight discrepancy may arise from:
- Higher-order terms in the expansion
- Vol-of-vol effects that modify the scaling
- The term structure of skew not being exactly power-law
Mechanism in stochastic volatility models: The \(1/\sqrt{T}\) decay arises from the interaction of two effects:
- Price-volatility correlation (\(\rho < 0\)): This generates skew. Over short horizons, a single volatility shock has a proportionally large effect on the return distribution's asymmetry
- Mean reversion of volatility (\(\kappa > 0\)): Over longer horizons, volatility reverts to its mean, diluting the impact of the current price-vol correlation. The effective skew contribution of any single shock decays as the volatility mean-reverts
Mathematically, the skew scales as \(\rho \xi / \sqrt{T}\) to leading order in stochastic volatility expansions, where \(\rho\) is the correlation and \(\xi\) is the vol-of-vol. The \(1/\sqrt{T}\) factor reflects the central limit theorem: over longer periods, the accumulated skew from many small shocks averages out relative to the total variance.
Exercise 4. Using the Breeden–Litzenberger formula, the risk-neutral density is
Suppose the call price function is \(C(K) = e^{-rT}[\mu - K + \frac{1}{2}\sigma^2 K^{-1}]\) for a hypothetical model. Compute \(\partial^2 C / \partial K^2\) and interpret the shape of the implied risk-neutral density. How would this density differ from a lognormal?
Solution to Exercise 4
Computing the second derivative:
Given \(C(K) = e^{-rT}\left[\mu - K + \frac{1}{2}\sigma^2 K^{-1}\right]\):
First derivative:
Second derivative:
Applying the Breeden–Litzenberger formula:
Interpretation: The risk-neutral density is \(f^{\mathbb{Q}}(x) = \sigma^2 / x^3\) for \(x > 0\). This is a heavy-tailed, right-skewed density (a Pareto-type distribution), meaning:
- Heavy right tail: The density decays as \(x^{-3}\), which is much slower than the exponential decay of a lognormal distribution. This implies significantly higher probability of extreme upward moves
- No left tail concentration: Unlike equity distributions, there is no extra weight on the left tail
- Infinite variance: The \(x^{-3}\) tail implies that the second moment \(\mathbb{E}[S_T^2] = \int \sigma^2 x^{-1}\,dx\) diverges, unlike a lognormal which has all moments finite
Difference from lognormal: A lognormal density has the form \(f(x) = \frac{1}{x\sigma_{\ln}\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu_{\ln})^2}{2\sigma_{\ln}^2}\right)\), which decays faster than any power law. The hypothetical \(K^{-3}\) density has much heavier tails and would generate a strongly upward-sloping implied volatility smile for high strikes.
Note: This hypothetical call price function is illustrative and may not satisfy all no-arbitrage conditions (such as proper boundary behavior), but it demonstrates how the Breeden–Litzenberger formula extracts distributional information.
Exercise 5. Explain the difference between "sticky strike" and "sticky delta" smile dynamics. A trader observes that when the S&P 500 drops by 2%, the implied volatility of a fixed-strike 4000 put increases by 1.5 vol points, while the 25-delta put implied vol remains roughly unchanged. Which smile model is more consistent with this observation? What are the hedging implications of using the wrong model?
Solution to Exercise 5
Sticky strike: The implied volatility of a specific strike \(K\) remains constant as the underlying moves. The smile is "anchored" to absolute strike levels.
Sticky delta: The implied volatility for a specific delta level (e.g., 25-delta put) remains constant as the underlying moves. The smile is "anchored" to moneyness relative to the current forward.
Analysis of the observation:
When the S&P 500 drops by 2%:
- The fixed-strike 4000 put's implied vol increases by 1.5 vol points — this means the implied volatility at strike 4000 is not constant; it increased. This is inconsistent with sticky strike, which would predict no change
- The 25-delta put implied vol remains roughly unchanged — this means the smile, when parameterized by delta (i.e., relative moneyness), is stable
This observation is consistent with sticky delta (floating smile). As the spot drops, the 4000 strike becomes less out-of-the-money (closer to ATM), and under sticky delta, it "picks up" the higher implied vol associated with its new, less-OTM delta level. Meanwhile, the 25-delta put (which now corresponds to a lower absolute strike) retains the same implied vol.
Hedging implications of using the wrong model:
- Using sticky strike when reality is sticky delta: The trader underestimates how much implied vol changes when the spot moves. Delta hedges will be systematically wrong because the model misses the skew-induced delta adjustment. Specifically, the "skew delta" correction:
is zero under sticky strike but nonzero under sticky delta, leading to under-hedging of downside risk
- Using sticky delta when reality is sticky strike: The trader over-adjusts for smile dynamics, introducing unnecessary hedging activity and transaction costs
Exercise 6. The variance risk premium is defined as \(VRP = \mathbb{E}^{\mathbb{Q}}[\sigma^2] - \mathbb{E}^{\mathbb{P}}[\sigma^2]\). Given a 3-month variance swap rate of \((17\%)^2\) and an expected realized variance of \((14\%)^2\), compute the annualized \(VRP\). A trader sells the variance swap at the swap rate and realizes the actual variance. What is the expected profit in variance points? Why is this premium typically positive for equity indices?
Solution to Exercise 6
Computing the annualized VRP:
The annualized \(VRP\) is \(93\) variance points (or \(0.93\%\) in variance terms).
Expected profit for the variance swap seller:
The trader sells the variance swap at the swap rate \((17\%)^2 = 0.0289\) and pays out the realized variance. The expected P&L is:
In volatility points: the trader receives an implied vol of 17% but expects to pay only 14%, netting approximately 3 vol points of profit (in variance terms, 93 variance points for a 3-month horizon).
Why the premium is typically positive for equity indices:
- Insurance demand: Institutional investors (pension funds, endowments) systematically buy portfolio insurance via puts and variance swaps, pushing up the risk-neutral expected variance
- Leverage effect: Volatility spikes during market downturns, so variance swap sellers lose money precisely when broader portfolios are also losing — this systematic risk requires compensation
- Negatively correlated with consumption: High volatility periods coincide with recessions and declining consumption, making variance exposure a source of systematic risk
- Jump risk: The risk-neutral measure overweights crash scenarios relative to the physical measure, inflating the variance swap rate
Exercise 7. A stochastic volatility model must jointly explain the smile, skew, and term structure. Fill in the following mapping and give a brief justification for each entry:
| Surface feature | Model parameter |
|---|---|
| Negative equity skew | ? |
| Smile curvature (wings) | ? |
| Term structure shape | ? |
| ATM level | ? |
Using the Heston model parameters (\(\kappa\), \(\theta\), \(\xi\), \(\rho\), \(V_0\)), identify which parameter primarily controls each feature.
Solution to Exercise 7
The completed table with Heston model parameter identification:
| Surface feature | Model parameter | Justification |
|---|---|---|
| Negative equity skew | \(\rho\) (price-vol correlation) | A negative \(\rho\) means that when \(S\) drops, \(V\) tends to rise, making the risk-neutral distribution left-skewed and generating higher implied vol for low strikes |
| Smile curvature (wings) | \(\xi\) (vol-of-vol) | A larger \(\xi\) increases the variability of volatility, fattening both tails of the risk-neutral distribution and raising implied vol for both deep OTM puts and calls |
| Term structure shape | \(\kappa\) (mean-reversion speed) | Fast mean reversion (\(\kappa\) large) causes short-term vol to revert quickly to \(\theta\), flattening the term structure; slow mean reversion preserves the gap between \(V_0\) and \(\theta\) |
| ATM level | \(V_0\) and \(\theta\) (current and long-run variance) | \(V_0\) determines the short-term ATM level, while \(\theta\) sets the long-term ATM level; the ATM implied vol interpolates between \(\sqrt{V_0}\) and \(\sqrt{\theta}\) |
Detailed justifications:
-
\(\rho\) controls skew: In the Heston model, the instantaneous correlation between \(dW^S\) and \(dW^V\) is \(\rho\). When \(\rho < 0\), a downward move in \(S\) coincides with an upward move in \(V\), so the conditional distribution after a price decline has higher variance. This asymmetry generates negative skew in the implied volatility surface. Typical equity index values are \(\rho \in [-0.9, -0.5]\)
-
\(\xi\) controls curvature: The vol-of-vol parameter determines how much \(V_t\) fluctuates around its mean. Higher \(\xi\) leads to greater dispersion in realized volatility paths, which translates to fatter tails in the terminal distribution and more pronounced smile curvature (higher wings)
-
\(\kappa\) controls term structure: The mean-reversion speed governs how quickly \(V_t\) returns to \(\theta\). For short maturities, implied vol is near \(\sqrt{V_0}\); for long maturities, it approaches \(\sqrt{\theta}\). The rate of transition is controlled by \(\kappa\)
-
\(V_0\) and \(\theta\) control ATM level: The current variance \(V_0\) sets the starting point and dominates short-maturity ATM implied vol, while the long-run mean \(\theta\) determines the asymptotic ATM level